Calculus Examples

$y = \sqrt[4]{\frac{2 x}{4 x - 3}}$

Step 1

Rewrite $\sqrt[4]{\frac{2 x}{4 x - 3}}$ as $\frac{\sqrt[4]{2 x}}{\sqrt[4]{4 x - 3}}$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x}}{\sqrt[4]{4 x - 3}}]$

Step 2

Multiply $\frac{\sqrt[4]{2 x}}{\sqrt[4]{4 x - 3}}$ by $\frac{{\sqrt[4]{4 x - 3}}^{3}}{{\sqrt[4]{4 x - 3}}^{3}}$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x}}{\sqrt[4]{4 x - 3}} \cdot \frac{{\sqrt[4]{4 x - 3}}^{3}}{{\sqrt[4]{4 x - 3}}^{3}}]$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[4]{2 x}}{\sqrt[4]{4 x - 3}}$ by $\frac{{\sqrt[4]{4 x - 3}}^{3}}{{\sqrt[4]{4 x - 3}}^{3}}$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{\sqrt[4]{4 x - 3} {\sqrt[4]{4 x - 3}}^{3}}]$

Step 3.2

Raise $\sqrt[4]{4 x - 3}$ to the power of $1$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{\sqrt[4]{4 x - 3}}^{1} {\sqrt[4]{4 x - 3}}^{3}}]$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{\sqrt[4]{4 x - 3}}^{1 + 3}}]$

Step 3.4

Add $1$ and $3$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{\sqrt[4]{4 x - 3}}^{4}}]$

Step 3.5

Rewrite ${\sqrt[4]{4 x - 3}}^{4}$ as $4 x - 3$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{4 x - 3}$ as ${(4 x - 3)}^{\frac{1}{4}}$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{({(4 x - 3)}^{\frac{1}{4}})}^{4}}]$

Step 3.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{(4 x - 3)}^{\frac{1}{4} \cdot 4}}]$

Step 3.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{(4 x - 3)}^{\frac{4}{4}}}]$

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{(4 x - 3)}^{\frac{4}{4}}}]$

Step 3.5.4.2

Rewrite the expression.

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{{(4 x - 3)}^{1}}]$

Step 3.5.5

Simplify.

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} {\sqrt[4]{4 x - 3}}^{3}}{4 x - 3}]$

Step 4

Rewrite ${\sqrt[4]{4 x - 3}}^{3}$ as $\sqrt[4]{{(4 x - 3)}^{3}}$ .

$\frac{d}{d y} [\frac{\sqrt[4]{2 x} \sqrt[4]{{(4 x - 3)}^{3}}}{4 x - 3}]$

Step 5

Combine using the product rule for radicals.

$\frac{d}{d y} [\frac{\sqrt[4]{2 x {(4 x - 3)}^{3}}}{4 x - 3}]$

Step 6

Since $\frac{\sqrt[4]{2 x {(4 x - 3)}^{3}}}{4 x - 3}$ is constant with respect to $y$ , the derivative of $\frac{\sqrt[4]{2 x {(4 x - 3)}^{3}}}{4 x - 3}$ with respect to $y$ is $0$ .

$0$